Effect of dynamical friction on nonlinear energetic particle modes

A fully nonlinear model is developed for the bump-on-tail instability including the effects of dynamical friction (drag) and velocity space diffusion on the energetic particles driving the wave. The results show that drag provides a destabilizing effect on the nonlinear evolution of waves. Specifically, in the early nonlinear phase of the instability, the drag facilitates the explosive scenario of the wave evolution, leading to the creation of phase space holes and clumps that move away from the original eigenfrequency. Later in time, the electric field associated with a hole is found to be enhanced by the drag, whereas for a clump it is reduced. This leads to an asymmetry of the frequency evolution between holes and clumps. The combined effect of drag and diffusion produces a diverse range of nonlinear behaviors including hooked frequency chirping, undulating, and steady state regimes. An analytical model is presented, which explains the aforementioned diversity. A continuous production of hole-clump pairs in the absence of collisions is also observed